THE IMPACT OF LATENT RISK PREFERENCES ON VALUING THE
PRESERVATION OF THREATENED LYNX POPULATIONS IN POLAND
Anna Bartczaka, Petr Marielb, Susan Chiltonc and Jürgen Meyerhoffd
a
University of Warsaw, Faculty of Economic Sciences, Warsaw Ecological Economics Center, ul. Dluga 44/50,
00-241 Warsaw, Poland
b
University of the Basque Country, (UPV/EHU), Avda. Lehendakari Aguirre, 83. E48015 Bilbao, Spain
c
Newcastle University Business School, 5 Barrack Road, Newcastle NE4 4SE, UK
d
Technische Universität Berlin, Institute for Landscape and Environmental Planning Straße des 17. Juni 145, D -
10623 Berlin
Keywords: choice experiment, hybrid latent class model, lottery experiment, lynx
preservation, risk preferences
1
Abstract
A recent innovation in stated preference environmental valuation surveys has been to
acknowledge the inherent uncertainties surrounding the provision and delivery of
environmental goods and services resulting from an intervention and to incorporate these into
non-market survey designs. However, little is known about how respondents assimilate and
respond to such uncertainty, particularly in terms of how it affects their stated valuations. In
this paper, we focus on the impact of risk on stated choices. Individual risk preferences are
elicited through a standard incentivised multiple price list mechanism. The results from this
lottery and a choice experiment concerning the chances of survival of an endangered species
are jointly analysed in a latent variable model by assuming that the responses to both are
driven by the same preferences. We find that the latent risk preferences are linked to choices
of the business-as-usual option without further protection measures, which is the riskiest
option in terms of the survival of endangered species.
2
1. Introduction
Owing to limited knowledge on natural processes and the fact that environmental
improvements often involve very long time horizons, the outcomes can entail substantial
uncertainty. Stated preference environmental valuation studies increasingly recognise and
attempt to incorporate such uncertainty. For example, to deal with limited knowledge in the
case of species preservation, environmental outcomes are often presented to respondents as
uncertain per se. The outcomes are described as changes in the chances of a population’s
survival, most commonly defined as an increase in the number of species by a specific
percentage, in a certain period (see Richardson and Loomis, 2009). The other source of
uncertainty is related to delivery. Supply uncertainty in respect of long-term projects can arise
for a number of reasons such as the limited reliability of institutions responsible for project
implementation or possible changes to the political, social and economic environment (see
e.g. Rigby et al., 2011, Rolfe and Windle, 2010, Glenk and Colombo, 2011).
These approaches constitute a significant methodological advance on the past, where
environmental outcomes were – explicitly or implicitly – presented as certain. However, how
individuals respond to and assimilate this uncertainty, particularly in respect to how it affects
their values, remains an open question. If we, like other practitioners, are willing to assume
that respondents comprehend the environmental good sufficiently well to answer to our
surveys, then we can begin to address this additional issue.
In this paper, we focus on the impact of risk preferences on values, although we
acknowledge that many other factors must play a part. Risk preferences have been shown to
influence behaviour in a number of other domains where uncertainty is a key feature of future
outcomes, such as health protection, financial investments, job changes or driving behaviour
(i.e. Anderson and Mellor, 2008; Kimball et al., 2008; Hakes and Viscusi, 2007; Weber et al.,
2002). However, only a few studies have thus far tested whether risk preferences measured in
3
an experimental way are linked with real risky behaviour. Anderson and Mellor (2008), for
example, showed that individuals who are more risk-averse are less likely to smoke and more
likely to wear seatbelts. Elston et al. (2005) reported that full-time entrepreneurs are less riskaverse than non-entrepreneurs and that part-time entrepreneurs were more risk-averse than
non-entrepreneurs. Lusk and Coble (2005) found that risk preferences are significant
determinants of the acceptance of genetically modified food. Meanwhile, Olbrich et al. (2011)
reported in their study that adult farmers in Namibia self-selected themselves onto farms
according to their risk preferences (i.e. those with lower risk aversion were found on farms
with higher environmental risks). All these studies, however, considered private goods,
whereas in our study we focus on a public good such as species conservation.
The main objective of the present paper is to examine the impact, if any, of individual
risk preferences on estimated willingness to pay (WTP) for lynx preservation in Poland. The
first part of the study is a choice experiment (CE) designed to value the preservation of the
two main lynx populations in Poland: the Lowland population that occupies the north-eastern
part of the country and the Carpathian located in the south. Both populations are exposed to a
high risk of becoming extinct. Mostly, this is the result of the rapid growth of transport
infrastructure and insufficient protection programmes. The outcomes of the conservation
programmes presented in the CE were specified as uncertain, which reflects scientific reality.
The second part of the study is designed to elicit respondents’ risk preferences. In this part,
we utilise the standard multiple price listing originally proposed by Binswanger (1980) and
later modified by Holt and Laury (2002). The major advantages of lottery CEs are that they
are relatively easy to explain, transparent and incentive-compatible.
The econometric approach we use to analyse our data is based on hybrid choice
models, which have been developed over recent years, with key developments by Ben-Akiva
et al. (1999, 2002) and Bolduc et al. (2005). This approach uses latent variables (LVs), which
4
are functions of the socio-demographic variables and an error term, to explain unobserved
latent risk preferences. At the same time, in a separate measurement model, these LVs are
used to explain answers to follow-up questions, related to psychological constructs included
in the model. In our study, the LV represents latent risk preferences.
The key advantage of the hybrid choice class of models is the use of additional data to
improve the precision of estimations and to better represent possible heterogeneity, which can
be modelled in various ways. The number of applications of hybrid choice models across
various fields has increased notably in recent years. Applications from an environmental
valuation perspective, for example, have been presented by Hess and Beharry-Borg (2012)
and Hoyos et al. (2013). A methodological novelty of the present study arises from the
application of a new approach to link the two parts of the hybrid choice model. We estimate a
model that resembles a latent class (LC)–mixed logit model by allowing for the interaction of
the LV with a utility coefficient in each class. In previous studies such as Daly et al. (2012)
and Glerum et al. (2012), the LV was interacted with attribute coefficients, in Hoyos et al.
(2013) the LV was explanatory in a class allocation function of an LC model and Hess and
Stathopoulos (2013) used the LV to explain scale heterogeneity within the choice model.
The remainder of paper is organised as follows. Section 2 presents the situation facing
lynx in Poland, Section 3 describes the case study and its methodology. Section 4 contains the
main results and, finally, Section 5 draws conclusions on the application of the hybrid choice
model.
5
2. Situation of lynx in Poland
The Eurasian lynx (Lynx lynx) is the third largest predator in Europe after the brown
bear and the wolf. Poland is one of the few European countries where lynx have survived in
the wild. However, the number of Polish lynx living in the wild has decreased to a third over
the past 20 years and is estimated to be about 180–200 individuals in total (Jędrzejewski et al.,
2002; von Arx et al., 2004). Although lynx have officially been protected in Poland since
1995, little has been done thus far to ensure the longer-term survival of the species
(Niedziałkowska et al., 2006). In general, their current status in Poland is considered to be
‘near threatened’ according to the IUCN Red List of threatened species.
There are two main lynx populations in Poland: the Lowland population in the
northeast and the Carpathian population in the south of the country. Both populations live in
border regions and are part of two major populations of this species in Europe. The Polish
Carpathian population is larger in number and more widely distributed than the Lowland
population and it is estimated to be about 100 animals. Existing migration corridors allow for
the exchange of the Carpathian lynx between countries. Meanwhile, the Lowland lynx
population, estimated at about 60 animals, occupies a highly fragmented habitat 1. This group
is more isolated from the lynx populations in other countries. These factors contribute to a
higher risk of extinction of the Lowland lynx in comparison to the Carpathian population (von
Arx et al., 2004).
Niedziałkowska et al. (2006) identified the fragmentation of forest habitats as a major
threat for the survival of the lynx populations in Poland. Other threats to the lynx populations
occur as a result of current forest management such as the afforestation of open spaces and
failing to leave enough deadwood in forests (Schmidt, 2008). Such changes in forests disturb
the lynx’s hunting and living conditions. Additionally, game hunting and poaching by humans
1
In addition to the Lowland and Carpathian lynx populations and a few isolated individuals in the north of
Poland, a group of 12–15 lynx lives in central Poland in the Kampinowski National Park. The group is isolated
and cannot survive in the wild without human support.
6
cause food scarcity. If these impacts on habitat conditions continue, it is anticipated that the
Polish lynx population may be seriously threatened in the next decades (Niedziałkowska et
al., 2006).
3. Survey design and methodology
3.1. Survey structure
The valuation survey consisted of six parts. Part 1 presented general information
concerning forests in Poland and questions about respondents’ recreation patterns in forests.
Part 2 provided general information on the lynx population in Europe and a more detailed
description of the two lynx populations in Poland. This information included a physical
description of the lynx, its habits, place of occurrence and main threats. Then, Part 3 depicted
potential management actions that could increase the chances of survival of the two main lynx
populations in the country. Among those actions, the most important was to create corridors
and passes across roads and railway tracks enabling the lynx to migrate between forest
complexes. In Part 4, the choice sets were presented to respondents. Part 5 was devoted to the
experimental elicitation of respondents’ risk preferences. Part 6 consisted of debriefing
questions to identify protest responses and standard socio-economic questions.
3.2. CE design
The CE comprises three attributes: the status of both the Lowland and the Carpathian
lynx population in 20 years from now and the annual cost of the particular conservation
programme per person 2. Following consultation with forest biologists, instead of employing
the commonly used increase in the number of individuals as a measure of the improved
2
The design of the CE follows Lew et al. (2010) to some extent. They investigated the public’s preferences for
enhancements to the protection of the western stock of Steller sea lions (Eumetopias jubatus).
7
protection of endangered species, we decided to describe the status of the lynx populations in
terms of its chances of survival. This form of presentation better reflects the inherent
ecological uncertainty surrounding lynx survival even in the presence of intervention and is
also, arguably, easier for respondents to understand than presenting chances of survival in
percentages or showing different population sizes. The categories used were based on the
IUCN Red List of threatened species. To clarify the categories, the official terminology was
simplified slightly (see Table 1). The final category descriptions along with the current and
the predicted status for both lynx populations were agreed through consultation with experts
from the Institute of Nature Conservation and Mammal Research Institute at the Polish
Academy of Sciences.
Table 1. Levels of threat.
IUCN Red List*
Scale adapted for the CE
Critically Endangered - Extremely high risk of
extinction in the wild.
Critically threatened - Extremely high risk
of extinction in the wild
Endangered - High risk of extinction in the wild.
Highly threatened - High risk of extinction in
the wild.
Vulnerable - High risk of endangerment in the
wild.
Moderately threatened - Moderate risk of
extinction in the wild.
Near Threatened - Likely to become
endangered in the near future.
Low threat level - Low risk of extinction in the
wild.
Least Concern - Lowest risk. Does not qualify
for a more at risk category.
Stable - Negligible risk of extinction in the wild.
Note: * The IUCN Red List includes two additional categories: extinct in the wild and extinct, which were not
included in the valuation survey, as they were not seen to be necessary for the purpose of our study.
For the purposes of the CE, the future status of the Lowland population could take one
of five levels (from critically threatened to stable), while for the Carpathian population four
attribute levels were used (from highly threatened to stable). The payment vehicle was a tax
8
that would go to a special fund established for lynx protection in Poland. Table 2 shows the
full list of attribute levels in the experimental design.
Table 2. Attributes and levels in the CE.
Attributes
Levels [Coding]
critically threatened [5] (expected for businessas-usual, i.e., without additional protection
Lowland lynx population (Lowland)
measures), highly threatened [4], moderately
threatened [3], low threat level [2], stable [1]
highly threatened [4] (expected for business-as-
Carpathian lynx population (Carpa)
usual), moderately threatened [3], low threat
level [2], stable [1]
0 zł (business-as-usual), 15 zł, 50 zł, 90 zł, 150
Cost per person per year (Cost)
zł 3
The choice sets were created by using a Bayesian D-efficient design with fixed priors
using the NGene software. The priors were gained from using responses from focus group
participants. As the efficiency measure, the so-called S-estimate was used (Bliemer and Rose,
2011). The final design comprised 24 choice sets that were blocked into four subsets. Each set
comprised two policy options and a business-as-usual option. Each option described the effect
the protection measures would have on the lynx populations’ chances of survival in the future.
Additionally, the sets provided information about the current number of individuals of each
population. To illustrate the differences between the hypothetical threat levels, colours
following the idea of traffic lights were used to mark attribute levels. Each attribute level was
accompanied by a pictogram of a lynx coloured according to the threat level. Each respondent
faced seven choice sets in total, including one with a dominant alternative; the latter choice
sets were not used in the present analysis. An example choice set is presented in Figure 1.
3
The nominal exchange rate from February 2011: 1 € = 3.9 zł.
9
Figure 1. Example choice set
Programme A
Programme B
Programme C
No additional
protection measures
Additional
protection measures
Additional
protection measures
Expected results
in 20 years
Expected results
in 20 years
Expected results
in 20 years
CRITICALLY
STABLE
POPULATION
CRITICALLY
THREATENED
of extinction
Negligible risk
of extinction
Extremely high risk
of extinction
HIGHLY
THREATENED
HIGHLY
THREATENED
MODERATELY
THREATENED
High risk
of extinction
High risk
of extinction
Moderate risk
of extinction
0 zł
90 zł
90 zł
THREATENED
LOWLAND
LYNX POPULATION Extremely high risk
Current number:
60 animals
CARPATHIAN
LYNX POPULATION
Current number:
100 animals
Cost per person
per year
I prefer the most
3.3. Measuring risk preferences with a lottery choice task
Based on the Holt and Laury (2002) approach, respondents were presented with a
series of 10 paired lotteries. For each of the 10 decisions, they were asked to choose either
lottery A or lottery B. In each decision, lottery A was the safe choice and lottery B was the
risky option. The payoffs for the safe option were less variable than those for the risky one.
For both lotteries, in each successive row, the likelihood of receiving larger rewards
increased. For the first four decisions, the expected payoff for lottery A was higher than that
for lottery B, while for the next six decisions lottery B had the higher expected payoff. In the
last row, no uncertainty was assigned to payoffs. Following Anderson and Mellor (2008), we
presented payoffs that were three times higher than the Holt and Laury (2002) baseline
amounts. Table 3 shows the full set of decision tasks.
10
Table 3. Lottery choice experiment.
Decision
Lottery A
Lottery B
E(A) –E(B)
1
Receive 18zł if dice throw is 1.
Receive 14.50zł if dice throw is 2-10.
Receive 34.70zł if dice throw is 1.
Receive 0.90zł if dice throw is 2-10.
10.6
2
Receive 18zł if dice throw is 1-2.
Receive 14.50zł if dice throw is 3-10.
Receive 34.70zł if dice throw is 1-2.
Receive 0.90zł if dice throw is 3-10.
7.5
3
Receive 18zł if dice throw is 1-3.
Receive 14.50zł if dice throw is 4-10.
Receive 34.70zł if dice throw is 1-3.
Receive 0.90zł if dice throw is 4-10.
4.5
4
Receive 18zł if dice throw is 1-4.
Receive 14.50zł if dice throw is 5-10.
Receive 34.70zł if dice throw is 1-4.
Receive 0.90zł if dice throw is 5-10.
1.5
5
Receive 18zł if dice throw is 1-5.
Receive 14.50zł if dice throw is 6-10.
Receive 34.70zł if dice throw is 1-5.
Receive 0.90zł if dice throw is 6-10.
-1.6
6
Receive 18zł if dice throw is 1-6.
Receive 14.50zł if dice throw is 7-10.
Receive 34.70zł if dice throw is 1-6.
Receive 0.90zł if dice throw is 7-10.
-4,6
7
Receive 18zł if dice throw is 1-7.
Receive 14.50zł if dice throw is 8-10.
Receive 34.70zł if dice throw is 1-7.
Receive 0.90zł if dice throw is 8-10.
-7.6
8
Receive 18zł if dice throw is 1-8.
Receive 14.50zł if dice throw is 9-10.
Receive 34.70zł if dice throw is 1-8.
Receive 0.90zł if dice throw is 9-10.
-10.6
9
Receive 18zł if dice throw is 1-9.
Receive 14.50zł if dice throw is 10.
Receive 34.70zł if dice throw is 1-9.
Receive 0.90zł if dice throw is 10.
-13.7
10
Receive 18zł if dice throw is 1-10.
Receive 34.70zł if dice throw is 1-10.
-16.7
To incentivise respondents to make their choices carefully, of these 10 decisions, one
was randomly selected as binding by the roll of a 10-sided die. Then, a die was thrown again
to determine whether the individual received the high or low real monetary payoff from the
chosen lottery.
11
3.4. Econometric approach
To capture more realistically the choice processes, we incorporated the latent
characteristics of decision makers into the model by treating the observed indicators of the
latent characteristics as endogenous (Bolduc and Alvarez-Daziano, 2010; Yáñez et al., 2010).
The hybrid model used in this application was composed of two sets of structural equations
and a group of measurement relationships.
The first set of structural equations is represented by the utilities of alternative 𝑖 for
respondent 𝑛 in the choice occasion 𝑡 as:
𝑈𝑖𝑛𝑡 = 𝑉𝑖𝑛𝑡 + 𝜀𝑖𝑛𝑡 = 𝐴𝑆𝐶𝑖 + 𝛽 ′ 𝑥𝑖𝑛𝑡 + 𝜀𝑖𝑛𝑡 ,
(1)
where 𝑉𝑖𝑛𝑡 is a systematic component and 𝜀𝑖𝑛𝑡 is a random variable following an extreme
value type I distribution with location parameter 0 and scale parameter 1. The term 𝑉𝑖𝑛𝑡
depends on observable explanatory variables, which are usually attributes (𝑥𝑖𝑛𝑡 ) and the
vector of estimated attribute parameters 𝛽. In (1), ASCi is an alternative specific constant for
alternative 𝑖 normalised to zero for one of the J alternatives.
The standard LC model specification forms the basis of the developments in this
paper. Given membership of class 𝑐𝑠 , the probability of respondent 𝑛’s sequence of choices is
given by
𝑇
𝑛
Pr(𝑦𝑛𝑡 |𝑐𝑠 , 𝑥𝑛 ) = ∏𝑡=1
𝑐
exp (𝐴𝑆𝐶𝑖 𝑠 +𝛽𝑐′𝑠 𝑥𝑖𝑛𝑡 )
𝑐
𝐽
∑𝑗=1 exp (𝐴𝑆𝐶𝑖 𝑠 +𝛽𝑐′𝑠 𝑥𝑗𝑛𝑡 )
,
(2)
where 𝑦𝑛𝑡 is the sequence of choices over the 𝑇𝑛 choice occasions for respondent 𝑛. Equation
(2) is a product of MNL probabilities. If the probability of membership to an LC 𝑐𝑠 of
respondent 𝑛 is defined as 𝜋𝑛,𝑐𝑠 , the unconditional probability of a sequence of choices can be
derived by taking the expectation over all 𝐶 classes, that is
12
𝑇
𝑛
𝑃𝑛 = Pr(𝑦𝑛𝑡 |𝑥𝑛 ) = ∑𝐶𝑠=1 𝜋𝑛,𝑐𝑠 ∏𝑡=1
exp (𝐴𝑆𝐶𝑖 +𝛽𝑐′𝑠 𝑥𝑖𝑛𝑡 )
𝐽
∑𝑗=1 exp (𝐴𝑆𝐶𝑖 +𝛽𝑐′𝑠 𝑥𝑗𝑛𝑡 )
.
(3)
The class allocation probabilities 𝜋𝑛,𝑐𝑠 are usually modelled by using a logit structure, where
the utility of a class is a function of a constant 𝜇0,𝑠 , the socio-demographics of the respondent
(𝑆𝐷𝑛 ) and corresponding parameters (𝜆𝑠 ), that is
𝜋𝑛,𝑐𝑠 =
exp�𝜇0,𝑠 +𝜆′𝑠 𝑆𝐷𝑛 �
C
∑s=1 exp�𝜇0,𝑠 +𝜆′𝑠 𝑆𝐷𝑛 �
(4)
.
The additional constant 𝜇0,𝑠 and parameters 𝜆 are fixed to zero for one of the classes for
normalisation reasons.
As the next step, we used the answers provided by respondents to the lottery choices
reported in Table 3. These answers are, together with respondents’ actual choices, driven by
underlying risk preferences; nevertheless, they are not direct measures of them. The latent risk
preferences are therefore treated as LVs and the lottery choices are used as indicators in the
model. The structural equation for the LV is given by
𝐿𝑉𝑛 = 𝛾1 𝑍1𝑛 + 𝛾2 𝑍2𝑛 + ⋯ + 𝛾𝑚 𝑍𝑚𝑛 + 𝜔𝑛 ,
(5)
where 𝑍1𝑛 , 𝑍2𝑛 , … , 𝑍𝑚𝑛 are the specific socio-demographic variables and 𝜔𝑛 is a random
disturbance that is assumed to be normally distributed with a zero mean and standard
deviation 𝜎𝜔 .
The measurement equations use the lottery choices as dependent variables, and
therefore as indicators of individuals’ risk aversion. The ℓth indicator of all 𝐿 indicators (in
our case, 𝐿 = 9 as in the last 10th lottery, lottery B was chosen by all individuals) for
respondent 𝑛 is defined as
(6)
𝐼ℓ𝑛 = 𝑚(𝐿𝑉𝑛 , 𝜁) + 𝑣𝑛 ,
13
where the indicator 𝐼ℓ𝑛 is a function of 𝐿𝑉𝑛 and a vector of parameters 𝜁 . The responses to the
lottery choice are binary; therefore, the measurement equation for individual 𝑛 is modelled as
a binary logit model for the LV:
𝐼ℓ𝑛 = �
𝑖𝑓
𝑖𝑓
0
1
− ∞ < 𝐿𝑉𝑛 ≤ 𝜏ℓ
,
𝜏ℓ < 𝐿𝑉𝑛 ≤ ∞
(7)
where 𝜏ℓ are the thresholds that need to be estimated. The likelihood of a specific observed
value of 𝐼ℓ𝑛 is then given by
exp(𝜏 −𝜁ℓ 𝐿𝑉𝑛 )
exp(𝜏 −𝜁ℓ 𝐿𝑉𝑛 )
� +𝐼(𝐼ℓ𝑛 =1) �1 − 1+exp(𝜏ℓ −𝜁
�,
)
−𝜁
𝐿𝑉
ℓ
ℓ 𝑛
ℓ
ℓ 𝐿𝑉𝑛 )
𝐿𝐼ℓ𝑛 = 𝐼(𝐼ℓ𝑛 =0) �1+exp(𝜏ℓ
(8)
where ζℓ measures the impact of LVn in indicator Iℓn and τℓ is estimated as the threshold
parameter. In the present study, we used a novel approach to link the two parts of the model.
We estimated an LC model and allowed interaction of the LV defined in (5) with the
alternative specific constant of the business-as-usual option (𝐴𝑆𝐶𝑆𝑄 ) in each class in order to
analyse the effect of the LV on the riskiest alternative in terms of the survival of both lynx
populations.
𝑐
The terms 𝑉𝑖𝑛𝑡 of (1) corresponding to class 𝑐𝑠 are defined as
𝑐
𝑐
𝑐
𝑐
𝑐
𝑐
𝑐
𝑐
𝑐
𝑐
𝑠
𝑠
𝑠
𝑠
𝑉1𝑛𝑡
= �𝐴𝑆𝐶𝑆𝑄𝑠 + 𝛿 𝑐𝑠 𝐿𝑉𝑛 � + 𝛽𝐶𝑎𝑟𝑝𝑎
𝐶𝑎𝑟𝑝𝑎1𝑛𝑡 + 𝛽𝐿𝑜𝑤𝑙𝑎𝑛𝑑
𝐿𝑜𝑤𝑙𝑎𝑛𝑑1𝑛𝑡 + 𝛽𝐶𝑜𝑠𝑡
𝐶𝑜𝑠𝑡1𝑛𝑡 (9)
𝑐
𝑐
𝑠
= 𝐴𝑆𝐶2 𝑠
𝑉2𝑛𝑡
𝑐
𝑠
𝑉3𝑛𝑡
=
𝑠
𝑠
𝑠
+ 𝛽𝐶𝑎𝑟𝑝𝑎
𝐶𝑎𝑟𝑝𝑎2𝑛𝑡 + 𝛽𝐿𝑜𝑤𝑙𝑎𝑛𝑑
𝐿𝑜𝑤𝑙𝑎𝑛𝑑2𝑛𝑡 + 𝛽𝐶𝑜𝑠𝑡
𝐶𝑜𝑠𝑡2𝑛𝑡
𝑠
𝑠
𝑠
𝛽𝐶𝑎𝑟𝑝𝑎
𝐶𝑎𝑟𝑝𝑎3𝑛𝑡 + 𝛽𝐿𝑜𝑤𝑙𝑎𝑛𝑑
𝐿𝑜𝑤𝑙𝑎𝑛𝑑3𝑛𝑡 + 𝛽𝐶𝑜𝑠𝑡
𝐶𝑜𝑠𝑡3𝑛𝑡
where 𝐶𝑎𝑟𝑝𝑎, 𝐿𝑜𝑤𝑙𝑎𝑛𝑑 and 𝐶𝑜𝑠𝑡 are the choice attributes described in Table 2, 𝛿 is the
parameter corresponding to the LV, which is added to the constant term in the business-asusual option, and 𝛽 are the class-specific attribute parameters.
14
The estimation of the model involves maximising the joint likelihood of the observed
sequence of choices and observed answers to the lottery choices. The two components are
conditional on the given realisation of 𝐿𝑉𝑛 . Accordingly, the log-likelihood function of the
model is given by integration over 𝜔𝑛
9
𝐿𝐿(𝛽, 𝜇, 𝛾, 𝜇, 𝜁, 𝜏) = ∑𝑁
𝑛=1 𝑙𝑛 ∫𝜔(𝑃𝑛 ∏ℓ=1 𝐿𝐼ℓ𝑛 ) 𝑔(𝜔)𝑑𝜔,
(10)
where 𝑃𝑛 is defined in (3), with class allocation probabilities 𝜋𝑛,𝑐𝑠 defined in (4), 𝐿𝐼ℓ𝑛 is
defined in (8) for ℓ = 1,2, … ,9. The joint likelihood function (10) depends on the parameters of
𝐶𝑠
𝐶𝑠
𝐶𝑠
the utility functions 𝛽 = (𝐴𝑆𝐶1𝐶𝑠 , 𝐴𝑆𝐶2𝐶𝑠 , 𝛽𝐶𝑎𝑟𝑝𝑎
, 𝛽𝐿𝑜𝑤𝑙𝑎𝑛𝑑
, 𝛽𝐶𝑜𝑠𝑡
), the parameters µ = (µ0,s ) and
𝑠
(𝜆1𝑠 , 𝜆2𝑠 , … , 𝜆𝐾
) contain the parameters used in the allocation probabilities defined in (4),
γ = (γ0 , γ1, γ2,…, γm ) contain the parameters for the socio-demographic interactions in the LV
specification defined in (5), and ζ = (ζ1 , ζ2 , … , ζ10 ) and τ = (τ1, , τ2 , … , τ10 ) contain the
parameters defined in (6) and (7). We follow the Bolduc normalisation by setting σω = 1. All
model components were estimated simultaneously by using PythonBiogeme (Bierlaire, 2003,
2008).
3.5. Data
Survey respondents were selected from Warsaw inhabitants. Ours was a quota sample
representative of the Warsaw population in terms of gender, age and education. The survey
was carried out in February 2011. Interviews were conducted by a professional polling agency
by using the computer-assisted personal interviewing system. In total, 300 questionnaires
were collected. The main survey was pretested in 50 face-to-face interviews with students
from the Faculty of Economics at the University of Warsaw.
15
Table 5. Descriptive statistics of the analysed sample.
Share
Women
Mean
Median
Min
Max
46
47
20
90
4,359
3,500
500
22,500
53%
Age
Education
-
Primary
8%
-
Secondary
49%
-
High
43%
Net monthly household income in zł
Individuals were excluded from the analysis if they chose the safe option for decision
10 in the lottery experiment or if they switched constantly between lottery A and lottery B for
all 10 decisions. We interpret this as not understanding the given instruction. Additionally,
respondents who always chose the most expensive alternative in the CE part were omitted.
We assumed that these individuals were protesting against some aspect of the survey. This
resulted in a final set of responses from 214 individuals corresponding to 1,284 observations
to be analysed.
4. Results
4.1 Risk preference elicitation
Of the analysed sample, 69% of respondents started with lottery A, then switched from
this option to lottery B just once and played this lottery thereafter. On the contrary, 31%
switched back from the risky lottery B to lottery A. Holt and Laury (2002), Lusk and Coble
(2005) and Anderson and Mellor (2008) also reported this kind of behaviour in their lottery
experiments, but the share of multiple switchers in their cases was lower (13%, 5% and 21%,
respectively). However, the first two surveys were conducted solely among students, while
only the Anderson and Mellor sample comprised mostly non-student adults. In the present
16
survey, respondents were recruited from the general public. Table 6 shows the share of “safe”
choices (lottery A) in the sample.
Table 6. Responses to the lottery
Decision
Share of “safe” choices (Lottery A)
Row 1
75%
Row 2
64%
Row 3
74%
Row 4
60%
Row 5
59%
Row 6
36%
Row 7
41%
Row 8
29%
Row 9
31%
4.2 Hybrid LC model results
Similar to a standard LC model framework, the first task when specifying a hybrid LC
model is to determine the number of classes. Table 7 reports the models’ goodness-of-fit
indicators together with the number of parameters. As expected, the log-likelihood decreases
as the number of classes increases. The BIC and CAIC indicate a solution with three classes,
while the AIC favours models with four classes. However, the AIC tends to overestimate the
number of classes and, moreover, there is consensus in the literature that parsimony is
preferable in modelling, especially in this complicated hybrid framework. Therefore, we
choose the LC model with three classes for further analysis.
Table 7. Goodness-of-fit criteria for different numbers of classes.
LogL
17
2 Classes
3 Classes
4 Classes
-1850.9
-1796.0
-1776.5
Number of parameters
39
50
61
AIC
3779.7
3692.1
3675.0
BIC
3980.9
3950.0
3989.6
CAIC
4019.9
4000.0
4050.6
Table 8 presents the estimations of the hybrid LC model and classical LC model. As
expected, there is little difference in the parameter estimates between these two estimations.
The hybrid model uses additional information for the estimation of the choice part model,
which allows for a richer interpretation.
The Cost coefficient in all three classes and two attribute parameters indicating an
improvement in lynx protection are significant at a level of 5% with the expected positive sign
in the second and third classes. Respondents in these two classes would be better off if
conservation measures leading to better lynx conservation were implemented, but at different
costs. However, lynx conservation attributes are not significant in the first class.
Table 8. Estimation results: choice model component.
LC Model
Class 1
Class 2
Class 3
0.06
0.06
0.88
Est.
rob. t-rat.
-4.21
-1.920 ***
Class prob.
ASC SQ
ASC A
Carpathian
Lowland
Cost
Est.
0.577
rob. t-rat.
0.60
-0.873
0.045
-0.367
-0.016 *
-1.18
0.16
-1.18
-1.94
Est.
-3.12 ***
rob. t-rat.
-8.41
*
**
**
***
1.85
2.20
2.47
-5.69
0.47
0.375
0.271
-0.035
7.720
0.455 ***
0.660 ***
-0.003 ***
0.86
5.03
7.57
-2.61
Hybrid LC Model
Class prob.
(median)
(25th, 75th
percentile)
ASC SQ
Coeff LV
Class 1
Class 2
Class 3
0.17
0.17
0.66
(0.14,0.23)
Est.
rob. t-rat.
-1.93
-11.900 *
2.30
15.200 **
(0.14,0.23)
Est.
rob. t-rat.
-3.93
-4.010 ***
-3.48
-3.270 ***
(0.53,0.72)
Est.
rob. t-rat.
-3.25
-1.640 ***
0.401
0.92
18
ASC A
Carpathian
Lowland
Cost
0.546
-0.038
0.103
-0.015 **
1.40
-0.12
0.48
-2.02
-0.057
0.520 *
0.251 *
-0.049 ***
-0.14
1.92
1.65
-3.11
0.066
0.531 ***
0.705 ***
-0.004 **
0.59
4.36
6.54
-2.39
The estimation of the allocation model (Table 9) showed no significant sociodemographics. However, in the hybrid LC model, apart from the constant variable, age was
also significant in the allocation function of class 3. The corresponding class probabilities
computed according to (4) are presented in Table 8. The highest probability is assigned to
class 3 in the two estimated models.
Table 9. Estimation results: Probability allocation functions..
LC Model
Class 2
𝜇0,2
𝜆𝐴𝑔𝑒,2
𝜆𝐹𝑒𝑚𝑎𝑙𝑒,2
𝜆𝐻𝑜𝑢𝑠𝑒ℎ𝑜𝑙𝑑𝑖𝑛𝑐𝑜𝑚𝑒,2
𝜆𝑈𝑛𝑖𝑣𝑒𝑟𝑠𝑖𝑡𝑦,2
Class 2
𝜇0,2
𝜆𝐴𝑔𝑒,2
𝜆𝐹𝑒𝑚𝑎𝑙𝑒,2
𝜆𝐻𝑜𝑢𝑠𝑒ℎ𝑜𝑙𝑑𝑖𝑛𝑐𝑜𝑚𝑒,2
𝜆𝑈𝑛𝑖𝑣𝑒𝑟𝑠𝑖𝑡𝑦,2
Est.
rob. t-rat.
1.610
1.32
0.018
0.10
-0.432
-0.77
0.271
0.20
-0.016
-0.02
Class 3
𝜇0,3
𝜆𝐴𝑔𝑒,3
𝜆𝐹𝑒𝑚𝑎𝑙𝑒,3
𝜆𝐻𝑜𝑢𝑠𝑒ℎ𝑜𝑙𝑑𝑖𝑛𝑐𝑜𝑚𝑒,3
𝜆𝑈𝑛𝑖𝑣𝑒𝑟𝑠𝑖𝑡𝑦,3
Est.
2.760 **
rob. t-rat.
2.53
-0.112
-0.69
-0.250
-0.47
0.190
0.14
-0.470
-0.80
Est.
rob. t-rat.
Hybrid LC Model
Est.
rob. t-rat.
1.490
1.25
-0.219
-1.08
-0.078
-0.11
-0.649
-0.74
-0.040
-0.06
Class 3
𝜇0,3
𝜆𝐴𝑔𝑒,3
𝜆𝐹𝑒𝑚𝑎𝑙𝑒,3
𝜆𝐻𝑜𝑢𝑠𝑒ℎ𝑜𝑙𝑑𝑖𝑛𝑐𝑜𝑚𝑒,3
𝜆𝑈𝑛𝑖𝑣𝑒𝑟𝑠𝑖𝑡𝑦,3
2.470 **
-0.277 *
2.43
-1.70
0.078
0.14
-0.212
-0.32
-0.572
-0.99
Table 10 presents estimation results of the structural and measurement equations and
confirms that the latent risk preferences influence respondents’ decision processes as the
19
impact of the LVs on the lottery choices was clearly significant for all nine latent risk
preference indicators (𝜁). Only household income of the four socio-demographic variables
included in the set of structural equations is significant (9), indicating that people with higher
household incomes have higher values of the LVs 4.
According to (7), a higher value of 𝐿𝑉𝑛 implies a lower probability of choosing the
safer lottery A (because the term 𝜏ℓ − 𝜁ℓ 𝐿𝑉𝑛 becomes lower) but a higher probability of
choosing the riskier lottery B. This result, therefore, indicates that respondents with higher
household incomes are more risk seeking than respondents with lower incomes. On the
contrary, for a given value of 𝐿𝑉𝑛 , the gradual decrease in the parameters 𝜁ℓ and 𝜏ℓ for the
sequence of lotteries 1–9 (Table 10) indicates a decrease in the probability of choosing the
safer lottery A for the benefit of the riskier choice of lottery B. This finding is in accordance
with the expected payoffs presented in Table 3.
The parameter 𝛿 corresponding to the LV added to the constant term in the SQ
alternative (8) is significant at 5% in two of the three classes (Table 8). In the first class, its
effect is positive and in the second, negative. Individuals with a high value for these constants
would choose (for the lynx population) the riskier SQ option with the higher probability. In
other words, the LV clearly affects respondents’ choices and therefore their WTP measures.
Table 10. Estimation results: structural and measurement equations.
Structural equation
Est.
rob. t-rat.
0.010
0.18
𝛾𝐹𝑒𝑚𝑎𝑙𝑒
-0.198
-0.86
𝛾𝑈𝑛𝑖𝑣𝑒𝑟𝑠𝑖𝑡𝑦
-0.056
𝛾𝐴𝑔𝑒
𝛾𝐻𝑜𝑢𝑠𝑒ℎ𝑜𝑙𝑑𝑖𝑛𝑐𝑜𝑚𝑒
0.435 **
3.47
-0.35
4
A low number of significant socio-demographics is a typical characteristic of hybrid choice models (Daly et al.,
2012).
20
Measurement equations
Est.
rob. t-rat.
𝜏1
3.42 **
2.52
𝜏2
1.55 *
1.79
𝜏3
2.43 ***
2.78
𝜏4
1.33
1.42
𝜏5
0.94
1.28
𝜏6
-0.65
-0.97
𝜏7
-0.38
-0.84
𝜏8
-1.14 **
-2.21
𝜏9
-0.95 ***
-2.63
Est.
rob. t-rat.
𝜁1
4.36 ***
4.02
𝜁2
3.34 ***
3.34
𝜁3
2.90 ***
4.08
𝜁4
3.20 ***
2.74
𝜁5
2.78 ***
5.12
𝜁6
2.48 ***
4.31
𝜁7
1.68 ***
3.97
𝜁8
1.81 ***
3.92
𝜁9
1.24 ***
3.71
In the next step, WTP measures were computed from the hybrid LC model estimates,
giving the implied monetary valuation of different changes in attribute levels. These values
are probability-weighted WTP values corresponding to the parameters presented in Table 8.
That is, for example, for the Carpathian population, the individual WTP values are computed
as:
𝑊𝑇𝑃 = 𝜋𝑛,𝑐1 �−
𝑐
1
𝛽𝐶𝑎𝑟𝑝𝑎
𝑐
1
𝛽𝐶𝑜𝑠𝑡
� + 𝜋𝑛,𝑐2 �−
𝑐
2
𝛽𝐶𝑎𝑟𝑝𝑎
𝑐
2
𝛽𝐶𝑜𝑠𝑡
� + 𝜋𝑛,𝑐3 �−
𝑐
3
𝛽𝐶𝑎𝑟𝑝𝑎
𝑐
3
𝛽𝐶𝑜𝑠𝑡
�
(11)
Table 11 shows, for the sample population of respondents, the distribution of the WTP values
for the two lynx populations. It also presents the WTP values computed by using the posterior
estimate of the LC probabilities defined as:
� 𝑛,𝑐𝑠
𝑃�𝑛|𝑐𝑠 𝜋
�
� 𝑛,𝑐
𝑠=1 𝑃𝑛|𝑐 𝜋
𝜋�𝑐𝑠 |𝑛 = ∑𝐶
𝑠
𝑠
,
(12)
where 𝑃�𝑛|𝑐𝑠 represents, for the given class assignment, the contribution of individual 𝑛 to the
likelihood through the joint probability of the sequence defined as:
21
𝑇𝑛
𝑃�𝑛|𝑐𝑠 = ∏𝑡=1
�𝑐′ 𝑥𝑖𝑛𝑡 )
� 𝑐𝑠 +𝛽
exp (𝐴𝑆𝐶
𝑖
𝑠
𝑐
𝐽
𝑠
�𝑐′ 𝑥𝑗𝑛𝑡 )
� +𝛽
∑
exp (𝐴𝑆𝐶
𝑗=1
𝑖
(13)
𝑠
According to the definition of the utility function (8), the constant (𝐴𝑆𝐶𝑖𝑐𝑠 ) for the
business-as-usual option is respondent-specific and a function of 𝐿𝑉𝑛 , which at the same time
depends on one socio-demographic variable (household income) and a random error term (5),
meaning that the constant follows a random distribution. In order to compute the posterior
estimate of the LC probabilities, we simulate the constant corresponding to the business-asusual option according to (5) and (9) by using 10,000 draws for the LV of each respondent,
combining the estimated parameter 𝛾𝐻𝑜𝑢𝑠𝑒ℎ𝑜𝑙𝑑𝑖𝑛𝑐𝑜𝑚𝑒 with the corresponding values of the
variable household income and adding generated random errors ω.
Table 11. Marginal WTP in zł.
25th percentile
Median
75th percentile
Using prior probabilities
Carpathian
75.43 zł
91.35 zł
100.83 zł
Lowland
98.02 zł
119.70 zł
132.61 zł
Carpathian
38.03 zł
134.39 zł
137.89 zł
Lowland
44.22 zł
178.29 zł
183.08 zł
Using posterior probabilities
As shown in Table 11, the posterior LC probabilities increase the spread of the two
distributions and shift them slightly to the right. The WTP for the Lowland population is
higher than that for the Carpathian population, suggesting that people prefer to invest more in
conservation programmes that protect the population at a higher risk of extinction.
Table 12 presents the same WTP values based on the posterior LC probabilities, but
this time they are simulated for four subgroups characterised by different age and household
22
income levels. Individuals assigned to the low (high) household income group are those with
a household income lower than the 25th (higher than the 75th) percentile. Similarly,
assignment to the younger and older age groups is based on the 25th and 75th percentiles of the
variable age.
Table 12. Distribution of the marginal WTP in zł for different subgroups.
Younger people
Low household income
Carpathian
25th
percentile
73.10 zł
Lowland
95.27 zł
High household income
135.95 zł
75th
percentile
137.91 zł
25th
percentile
86.59 zł
137.66 zł
75th
percentile
137.92 zł
180.48 zł
183.10 zł
111.98 zł
182.76 zł
183.12 zł
Median
Median
Older people
Low household income
Carpathian
25th
percentile
20.67 zł
Lowland
22.28 zł
High household income
128.83 zł
75th
percentile
137.85 zł
25th
percentile
27.09 zł
130.03 zł
75th
percentile
137.88 zł
170.95 zł
183.01 zł
28.23 zł
172.18 zł
183.06 zł
Median
Median
As expected, older people are less willing to invest in 20-year protection programme
than younger people. On the contrary, people with a higher household income are willing to
pay more than people with a low household income. An interesting result is that the
differences in WTP distributions are much higher between younger and older people than
between people with low and high household incomes. This finding means that people are
aware of the risk of lynx extinction in Poland and are willing to invest in conservation
programmes; nevertheless, their WTP increases slowly with household income but decreases
rapidly with age.
23
5. Conclusions
This paper extents our knowledge about the association between individual risk
preferences and investments in an environmental good with uncertain results. To our
knowledge this study has introduced two new elements to a choice experiment literature, the
use of an incentivized multiple price list to elicit risk preferences and the application of a
latent variable model to link the responses to the lottery and to the choice experiment. Using
this approach we analysed the role that latent risk preferences may play in people’s
preferences towards lynx protection in Poland. The results show that individuals’ latent risk
preferences are significantly related to the constant of the business-as-usual option, i.e. the
option without additional protection measures and a zero-price, influencing, therefore, the
probability that it will be chosen. It is also noteworthy that the incorporated LV is
significantly related to the variable household income. This finding is in line with other
findings in the literature, suggesting that risk preferences may be correlated with wealth (see
e.g. Rosenzweig and Binswanger, 1993; Liu, 2013) 5. At the same time, the LV provides a
strong explanation of respondents’ choices in the series of nine lotteries. Our results therefore
confirm the findings from other studies indicating that respondents’ choices are, apart from
the attributes of the alternatives, related to their latent risk preferences.
This paper also raises the possibility that the choice of the business-as-usual option
may be linked to fundamental risk preferences, even if exacerbated by psychological
influences such as framing or anchoring effects (Tversky and Kahneman, 1974). Often, in CE
studies of changes in environmental management, the business-as-usual option without further
measures is the riskiest of the presented alternatives. The other alternatives are usually
programmes aimed at improving the current environmental situation. Therefore, it may be
prudent to elicit individual risk preferences as a possible explanatory variable in order to
5
However, the literature on whether risk preferences vary with wealth level is inconclusive (see Cardenas and
Carpenter, 2008).
24
estimate the impact on estimated WTP. The methodological novelty of this study arises from
developing a new approach to linking the LVs to the choice model part. We estimate the
hybrid model, which resembles an LC model but allows for the interaction of the LV with an
attribute (in this case, the ASC of the business-as-usual option) in each class.
The combination of a lottery and choice experiment seems to be a productive
combination. The Holt and Laury lottery, however, was designed to capture risk preferences
in the financial domain. While it has been shown in the literature that risk preferences elicited
using this approach are also meaningful in other domains, for example for predicting health
behaviours (Andersen and Mellor, 2010), lotteries aiming directly at environmental risks
might be more suitable as a determinant of choices among alternatives with uncertain
environmental outcomes. In addition, the application of the LV approach is still new to
environmental valuation. So far the results indicate that these models give richer insights into
what determines choice among the offered alternatives. On the other hand, the WTP estimates
as the final outcome are not strongly affected by the use of the LV model. Whether the results
are specific to our data can only be answered by future studies.
25
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